If you look into zombie lore a bit, there are two types of zombie - the first likes brains, the second likes living flesh.

Either way, the zombie itself is dead - eating living flesh or brain is a survival mechanism. After all, if you could eat dead flesh, why not just eat your fellow zombies?

However, it's also worth pointing out that zombies don't have the cognitive function necessary to decide to go to the store for brains. Zombies are probably less intelligent than dogs - have you ever seen zombie who knew "Sit" or "play dead?"